Sho Sugiura

Thermal pure quantum states at finite temperature.

An equilibrium state can be represented by a pure quantum state, which we call a thermal pure quantum (TPQ) state. We propose a new TPQ state and a simple method of obtaining it. A single realization of the TPQ state suffices for calculating all statistical-mechanical properties, including correlation functions and genuine thermodynamic variables, of a quantum system at finite temperature.

Canonical Thermal Pure Quantum State

A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.

Thermal pure quantum states of many-particle systems

We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1d) chain and on a two-dimensional (2d) triangular lattice. For the 1d chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential and the on-site interaction. For the 2d triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ (cTPQ) state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.

Universality in volume-law entanglement of scrambled pure quantum states

A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum correlations break the correspondence and mandate a correction to this simple volume law. The elucidation of the size dependence of the entanglement entropy is thus essentially important in linking quantum physics with thermodynamics. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing equilibrium. We numerically find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., energy eigenstates of non-integrable models and states after quantum quenches. Our formula is exploited as diagnostics for chaotic systems; it can distinguish integrable models from non-integrable models and many-body localization phases from chaotic phases.The entanglement in a quantum system between a small region and the surrounding environment contains details about the whole state. Nakagawa et al. find a formula for the entanglement entropy of a class of thermal-like states and show that it can be applied more broadly to identify equilibrating states.

Typicality and Ergodicity

“The principle of equal a priori probability (PEPP)” is the postulate that all the realizable microstates appear in the same probability. In this chapter, I review that we can understand the PEPP from the viewpoint of “typicality,” which states that most of microstates in ensemble can represent the corresponding equilibrium state. In classical statistical mechanics, an observation of thermodynamic fluctuation leads to a notion of the typicality in a weak sense (weak typicality). Moreover, in quantum statistical mechanics, a superposition of many energy eigenstates makes the typicality hold in much stronger sense (strong typicality).

Application to Numerical Calculation

The TPQ formulation is not only interesting for the foundation of statistical mechanics, but also useful for practical applications. I compare this formulation to other numerical methods and explain its advantage in 2D frustrated spin and electron systems. As an illustration, I show the numerical results for antiferromagnetic Heisenberg model on the Kagome lattice.

Introduction to Thermal Pure Quantum State Formulation of Statistical Mechanics

Quantum statistical mechanics is the theory which gives the thermodynamic predictions from quantum mechanics. A thermal equilibrium state is conventionally described by a mixture of pure quantum state. However, a single realization of pure quantum states can also represent the thermal equilibrium. I call this pure state a thermal pure quantum state, and establish a formulation of statistical mechanics based on it.

Relation Among TPQ States

In this chapter, I show transformation relations among the TPQ states which correspond to different ensembles: microcanonical, canonical, and grandcanonical ensembles. These relations make the TPQ formulation possible to apply to numerical calculations. I also show a similar relation which is more useful to the numerical calculations.

Equilibrium State and Entanglement

In this chapter, I see some aspects of the TPQ formulation. Regarding macroscopic quantities, the TPQ states behave in the same way as the Gibbs states do. However, these two states are different when we look at the representation of the fluctuation and the quantum entanglement.

Microcanonical Thermal Pure Quantum State

In the microcanonical ensemble, an equilibrium state is specified by energy. In this case, the corresponding thermal pureTPQ state is a microcanonical TPQ (mTPQ) state. I will introduce the mTPQ state with an efficient construction method for practical applications.